•Novel numerical vortex-based scheme for incompressible Euler flow.•Vorticity-Bernoulli-pressure formulation.•Results compare well with unsteady analytical solutions.•Second-order accurate in space and accurate in long-time simulations. A novel numerical formulation and time-integration method for the incompressible Euler flow equations is presented using the vorticity–Bernoulli-pressure system. Lagrangian advection of vorticity-carrying particles is adopted from classical vortex methods that are known to give accurate results for vortex-dominated flows and allow larger time steps than explicit Eulerian schemes. Solving only one scalar Poisson equation per time step instead of a vector-valued equation in vortex methods makes the new scheme theoretically more efficient on large computational grids. The equation of motion in rotation form is integrated in time under certain approximation assumptions and solved explicitly for the flow velocity. The velocity is eliminated as an unknown in the continuity equation, which is efficiently solved for the Bernoulli pressure using the predicted vorticity field obtained by Lagrangian advection. The Bernoulli pressure is efficiently obtained from the Poisson equation in finite volume formulation. An estimated velocity field is calculated from the equation of motion. To effectively counterbalance the discretization errors made by the prediction, in a corrector step the kinematic relation between vorticity and estimated velocity is reformulated and solved in a Lagrangian framework as well. The velocity field is corrected at the end of the time step. Two-dimensional cases on Cartesian grids are considered. Numerical results for three flow cases demonstrate the locally smooth and accurate behavior and the second-order spatial accuracy of the solution. The new scheme produces significantly less artificial vortex distortion than the classical vortex method. The favorable properties of the solutions with respect to accuracy and long-time stability of the overall kinetic energy and enstrophy are also emphasized.